Moduli spaces and braid monodromy types of bidouble covers of the quadric

Catanese, Fabrizio¹ ; Lönne, Michael¹ ; Wajnryb, Bronislaw²

¹ Mathematisches Institut, Lehrstuhl Mathematik VIII, Universität Bayreuth, Universitätsstraße 30, D-95447 Bayreuth, Germany
² Department of Mathematics, Rzeszów University of Technology, ul W Pola 2, 35-959 Rzeszów, Poland

Geometry & topology, Tome 15 (2011) no. 1, pp. 351-396.

Résumé

Bidouble covers $\pi :S\to Q:={\mathbb{P}}^1\times {\mathbb{P}}^1$ of the quadric are parametrized by connected families depending on four positive integers $a,b,c,d$. In the special case where $b=d$ we call them $abc$–surfaces.

Such a Galois covering $\pi$ admits a small perturbation yielding a general $4$–tuple covering of $Q$ with branch curve $\Delta$, and a natural Lefschetz fibration obtained from a small perturbation of the composition $p_1\circ \pi$.

We prove a more general result implying that the braid monodromy factorization corresponding to $\Delta$ determines the three integers $a,b,c$ in the case of $abc$–surfaces. We introduce a new method in order to distinguish factorizations which are not stably equivalent.

This result is in sharp contrast with a previous result of the first and third author, showing that the mapping class group factorizations corresponding to the respective natural Lefschetz pencils are equivalent for $abc$–surfaces with the same values of $a+c,b$. This result hints at the possibility that $abc$–surfaces with fixed values of $a+c,b$, although diffeomorphic but not deformation equivalent, might be not canonically symplectomorphic.

Détail

DOI : 10.2140/gt.2011.15.351

Keywords: algebraic surface, moduli space, braid monodromy, equivalence of factorizations, symplectomorphism, bidouble cover, Lefschetz pencil

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